Generally, a resonant sensor is characterized by a resonance frequency which depends mainly on the mass of the movable part of the sensor, the geometric parameters of the mechanical parts of the sensor and the physical parameters of the materials forming the various parts of the sensor, as well as the quality factor dependent on the energy losses of the resonant sensor. The dependency of the resonance frequency with the aforementioned physical parameters, thereby makes it possible to carry out a measurement of an exterior perturbation of one of these parameters (for example a variation of mass, an acceleration, a pressure, etc.).
Thus in applications of this type, a resonant mechanical element, that may for example be a clamped/clamped or clamped free nano-beam, is conventionally equipped with at least one electrode for actuation EA making it possible to apply a drive voltage, such as illustrated in FIG. 1 and which represents a clamped beam P.
The electrostatic loading generated by the drive voltage may then be written
      F    act    =            1      2        ⁢          ɛ      0        ⁢    S    ⁢                  V        2                              (                      g            -            x                    )                2            
where S is the facing surface area, g the gap between the drive electrode and the beam, V the actuation voltage comprising a DC component and an AC component such that: V=Vdc+Vac·Cos Ωt. The DC voltage Vdc bends the beam statically whereas the harmonic voltage Vac·Cos Ωt bends it dynamically.
Conventionally the resonator is caused to vibrate at its fundamental bending frequency so as to obtain the maximum of amplitude. By virtue of an inverse electrical transduction, it is then possible to determine the resonance frequency of the device from reading the signal.
In the case for example of a mass sensor that may be the one shown diagrammatically in FIG. 1, under the effect of an added mass or of an axial load, the resonance frequency is shifted from the frequency f0, to the frequency fres, as illustrated in FIG. 2. The measurement of the frequency shift δf is equal to f0−fres with a resonance frequency measured fres, after detection that may be of capacitive type.
In a conventional manner, the amplitude of actuation of the resonator is maintained below a so-called critical amplitude, beyond which the vibration regime becomes nonlinear.
Nonetheless, to improve the performance of a resonator, it is possible to seek to obtain the largest possible amplitude of actuation of the resonator. The amplitude then exceeds a threshold value corresponding to the critical amplitude beyond which the vibration regime becomes non-linear, causing the occurrence of hysteresis phenomena. The occurrence of this non-linear regime is notably described in the article by N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology, vol. 20, p. 275501, 2009 or in patent application EP 2 365 282.
FIG. 3 illustrates this principle, within the framework of the frequency response of a nano-resonator of beam type with a so-called softening non-linear device (the peak is deviated towards the low frequencies). The principle would be identical with a perfectly linear or non-linear response of a so-called stiffening device (peak deviated towards the right). In the absence, for example, of any additional mass to be detected, the response of the nano-resonator as a function of frequency for amplitude values beyond the linearity threshold has the behaviour of the initial response curve F0.
Under the effect of a very meagre added mass, the response curve of the nano-resonator is shifted towards the low frequencies to give the curve FM. Conventional frequency measurements consist in detecting this shift ΔΩ and in measuring it. However, this shift becomes very small and difficult to distinguish from measurement noise for very small masses.
It is theoretically possible to improve the frequency sensitivity by decreasing the sizes and/or by increasing the signal-to-noise ratio, that is to say by actuating the resonators in a more significant manner. However, under these conditions, the nano-resonators have a very strongly non-linear behaviour, a source of instabilities and of mixing of low and high frequency noise that are liable to degrade the reliability and precision of the measurements by frequency shifting as described in the article by V. Kaajakari, J. K. Koskinen, and T. Mattila, “Phase noise in capacitively coupled micromechanical oscillators,” IEEE transactions on ultrasonics ferroelectrics, and frequency drive, vol. 52, no. 12, pp. 2322-31, December 2005.
Another route for improving the sensitivity of resonant sensors consists in defining alternative detection principles based on the exploitation of non-linear phenomena. Several studies have already been described in the literature, which seek for example to amplify the resonator's response amplitude by means of internal or parametric resonances and notably in the articles by: W. Zhang and K. L. Turner, “Application of parametric resonance amplification in a single-crystal silicon micro-oscillator based mass sensor,” Sensors and Actuators A: Physical, vol. 122, no. 1, pp. 23-30, July 2005 or by M. I. Younis and F. Alsaleem, “Exploration of New Concepts for Structures Based on Nonlinear Phenomena,” Journal Of Computational And Nonlinear Dynamics, vol. 4(2), 021010, 2009. But these resonances exist only when the resonators have very particular geometries and excitations.
An alternative also disclosed within the field consists in using the jumps in amplitude in the neighbourhood of singular operating points and is illustrated by virtue of FIG. 4. More precisely, this entails causing the resonator to vibrate without extra mass at a fixed frequency Ωop slightly lower than that of the limit point Alim of the response curve F0 (without perturbation). More precisely, this point Alim corresponds to a bifurcation point, corresponding to a change of increase and of decrease of the frequency.
The trend of this response curve is characteristic of a non-linear behaviour and, at the chosen excitation frequency Ωop, it possesses two stable operating points A1 and A2.
FIG. 4 thus highlights a so-called unstable frequency band BINS in which two stable amplitudes can correspond to a single frequency: for example the amplitudes A1 and A2 to the frequency Ωop, whereas below a certain frequency value, it is possible to define a so-called stable frequency band BS in which there is indeed correspondence between a frequency and a single stable amplitude.
In this configuration, when a mass is added to the resonator, the response curve is shifted towards the curve FM. Given that this new response curve with added mass possesses only a single operating point B at the frequency Ωop, an abrupt jump in amplitude from A1 to B occurs, as described in the article by V. Kumar, S. Member, Y. Yang, S. Member, G. T. Chiu, and J. F. Rhoads, “Modeling, Analysis, and Experimental Validation of a Bifurcation-Based Microsensor,” Journal of Microelectromechanical Systems, vol. 21, no. 3, pp. 549-558, 2012.
In contradistinction to frequency detection based on ΔΩ, this jump is all the larger as the extra mass is small, thereby rendering this technique particularly beneficial. Moreover, the detection threshold in terms of mass can be tailored with the value of the frequency Ωop. It is thus possible to quantify the mass deposited by virtue of the amplitude of the jump in amplitude, but also simply to detect the presence or otherwise of the mass, and to count the number of particles which have deposited.
Nonetheless, once mass detection has been effected, the particle-free nano-beam must be able to regain its initial state, that is to say regain the state A1. In the converse case, if instead of dropping back to its operating point A1, the nano-beam jumps from its state B to the state A2, it becomes, in this case, difficult to again detect an appreciable amplitude variation. A reinitialization phase becomes necessary in order to carry out new sensitive measurements.